Problem 976 - JEE Physics

Question

Cross-sectional area o if wire of length \(L\) is \(A\). Young's modulus of material is \(\mathrm{Y}\). If this wire acts as a spring what is the value of force constant? (A) (YA/L) (B) (YA/2L) (C) (2YA/L) (D) ( \(\mathrm{YL} / \mathrm{A})\)

Step-by-Step Solution

Answer

The correct value of the force constant is (A) (YA/L).

Step 1: Recall Hooke's Law

Hooke's Law states that the force acting on a spring is proportional to the displacement of the spring from its equilibrium position. Mathematically, it is given by: \(F = -kx\) Where F is the force, k is the spring constant (or force constant), and x is the displacement of the spring from its equilibrium position.

Step 2: Relate stress and strain using Young's modulus

We have to relate the force F to the length L and cross-sectional area A of the wire. To do this, we use the definition of Young's modulus, which states the relationship between stress and strain: \(Y = \dfrac{\text{stress}}{\text{strain}}\) Stress is defined as the force per unit area, while strain is defined as the change in length per unit length. We can write these definitions mathematically as: \(\text{stress} = \dfrac{F}{A}\) and \(\text{strain} = \dfrac{\Delta L}{L}\)

Step 3: Find the force constant using Hooke's Law and Young's modulus

Now, we can substitute the expressions for stress and strain in the Young's modulus equation: \(Y = \dfrac{\dfrac{F}{A}}{\dfrac{\Delta L}{L}}\) Next, we solve this equation for the force F: \(F = \dfrac{YA}{L}\Delta L\) By comparing this equation with Hooke's Law, we can see that the force constant, k, is given by: \(k = \dfrac{YA}{L}\)

Step 4: Find the value of the force constant in the given options

Now, we can find the force constant value in the given options: (A) (YA/L) (B) (YA/2L) (C) (2YA/L) (D) (YL/A) We derived the force constant, k, as (YA/L), which matches option (A). Therefore, the correct answer is: (A) (YA/L)

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in physics that helps describe the behavior of springs and elastic materials. It states that the force needed to extend or compress a spring by a certain distance is proportional to that distance. In simple terms, the more you stretch or compress a spring, the more force you need to do that. Mathematically, Hooke's Law is described as:\( F = -kx \)where:

• \( F \) is the force applied to the spring.
• \( k \) is the force constant (also known as the spring constant).
• \( x \) is the displacement of the spring from its rest position.

The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement. The principle is widely applicable in various fields, including engineering and material science, helping us understand how different materials respond to applied forces. It gives a basic, yet powerful tool, to design and analyze applications involving springs or elastic components.

Force Constant

The force constant, often denoted as \( k \), is a measure of the stiffness of a spring or an elastic object. It tells you how difficult it is to either stretch or compress the object. The greater the force constant, the stiffer the spring, and consequently, the more force required to change its shape.In our problem, we derived the force constant using Young's modulus by relating it to the wire acting like a spring:\( k = \dfrac{YA}{L} \)Where:

• \( Y \) is Young's modulus, a property of the material.
• \( A \) is the cross-sectional area of the wire.
• \( L \) is the original length of the wire.

This formula emphasizes how material properties and physical dimensions influence the spring behavior of the wire. Materials with a high Young's modulus and smaller cross-sectional areas tend to have larger force constants, indicating stiffer behavior. Understanding the force constant is essential for designing systems where elastic deformation is a key consideration.

Stress and Strain Relationship

Stress and strain are two critical concepts that describe how materials deform under force. Stress is the force exerted per unit area on an object, while strain is the deformation or change in length relative to the object's original length.The relationship between stress and strain for a material is often characterized by Young's modulus (\( Y \)), which is a measure of the elasticity of a material. It is defined as:\[ Y = \dfrac{\text{stress}}{\text{strain}} \]Where:

• \( \text{stress} = \dfrac{F}{A} \), with \( F \) being the applied force and \( A \) the cross-sectional area.
• \( \text{strain} = \dfrac{\Delta L}{L} \), where \( \Delta L \) is the change in length, and \( L \) is the original length.

This relationship helps us understand how resistant a material is to being deformed elastically (non-permanently) when a force is applied. Materials with high Young's moduli are less prone to deformation and are thus stiffer. Analyzing stress and strain helps engineers select appropriate materials for different applications, ensuring components can withstand applied forces without permanent damage.